3.75 \(\int \sqrt{x} \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=52 \[ \frac{2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}-\frac{4 b \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}} \]

[Out]

(-4*b*(b*x + c*x^2)^(3/2))/(15*c^2*x^(3/2)) + (2*(b*x + c*x^2)^(3/2))/(5*c*Sqrt[
x])

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Rubi [A]  time = 0.0570127, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}-\frac{4 b \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*Sqrt[b*x + c*x^2],x]

[Out]

(-4*b*(b*x + c*x^2)^(3/2))/(15*c^2*x^(3/2)) + (2*(b*x + c*x^2)^(3/2))/(5*c*Sqrt[
x])

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Rubi in Sympy [A]  time = 5.9992, size = 46, normalized size = 0.88 \[ - \frac{4 b \left (b x + c x^{2}\right )^{\frac{3}{2}}}{15 c^{2} x^{\frac{3}{2}}} + \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{5 c \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)*(c*x**2+b*x)**(1/2),x)

[Out]

-4*b*(b*x + c*x**2)**(3/2)/(15*c**2*x**(3/2)) + 2*(b*x + c*x**2)**(3/2)/(5*c*sqr
t(x))

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Mathematica [A]  time = 0.024281, size = 41, normalized size = 0.79 \[ \frac{2 \sqrt{x (b+c x)} \left (-2 b^2+b c x+3 c^2 x^2\right )}{15 c^2 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*Sqrt[b*x + c*x^2],x]

[Out]

(2*Sqrt[x*(b + c*x)]*(-2*b^2 + b*c*x + 3*c^2*x^2))/(15*c^2*Sqrt[x])

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Maple [A]  time = 0.005, size = 33, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -3\,cx+2\,b \right ) }{15\,{c}^{2}}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)*(c*x^2+b*x)^(1/2),x)

[Out]

-2/15*(c*x+b)*(-3*c*x+2*b)*(c*x^2+b*x)^(1/2)/c^2/x^(1/2)

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Maxima [A]  time = 0.723939, size = 41, normalized size = 0.79 \[ \frac{2 \,{\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} \sqrt{c x + b}}{15 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*sqrt(x),x, algorithm="maxima")

[Out]

2/15*(3*c^2*x^2 + b*c*x - 2*b^2)*sqrt(c*x + b)/c^2

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Fricas [A]  time = 0.236252, size = 70, normalized size = 1.35 \[ \frac{2 \,{\left (3 \, c^{3} x^{4} + 4 \, b c^{2} x^{3} - b^{2} c x^{2} - 2 \, b^{3} x\right )}}{15 \, \sqrt{c x^{2} + b x} c^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*sqrt(x),x, algorithm="fricas")

[Out]

2/15*(3*c^3*x^4 + 4*b*c^2*x^3 - b^2*c*x^2 - 2*b^3*x)/(sqrt(c*x^2 + b*x)*c^2*sqrt
(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x} \sqrt{x \left (b + c x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)*(c*x**2+b*x)**(1/2),x)

[Out]

Integral(sqrt(x)*sqrt(x*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.210571, size = 46, normalized size = 0.88 \[ \frac{4 \, b^{\frac{5}{2}}}{15 \, c^{2}} + \frac{2 \,{\left (3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b\right )}}{15 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*sqrt(x),x, algorithm="giac")

[Out]

4/15*b^(5/2)/c^2 + 2/15*(3*(c*x + b)^(5/2) - 5*(c*x + b)^(3/2)*b)/c^2